Growth models of cultures with two lipid phases. I. Substrate dissolved in dispersed phase

Mathematical models which can be used to describe batch growth in fermentations with two liquid phases are developed for systems in which the growth limiting substrate is dissolved the dispersed liquid phase. In the models, the possibilities of growth occurring at the surface of the dispersed phase and in the continuous phase are considered. It is assumed that the composition of the dispersed phase is such that substrate utilization from it causes little of no change, in interfacial area. Three special cases are examined. In the first, it is assumed that all growth occurs at the surface of the dispersed phase. In the second and third, it is assumed that growth occurs both at the interface and in the continuous phase. The second case assumes that substrate equilibrium is continuously established between the two phases while the third assumes substrate consumption in the continuous phase is limited by rate of transport of substrate to that phase. Comparison of the first model with available experimental data shows good agreement between model and data.     

Growth models of cultures with two liquid phases. II. Pure substrate in dispersed phase

Mathematical models which can be used to describe batch frowth in fermentations with two liquid phases are developed for systems in which the growth limiting substrate is the dispersed liquid phase. Three special cases are considered assuming pure substrate in the dispersed phase and a decreasing interfacial area due to substrate consumption. In the first, it is assumed that all growth occurs at the surface of the dispersed phase. In the second and third growth occurs at the interface and in the continuous phase. The second case assumes substrate equilibrium between the two phases while the third assumes substrate consumption in the continuous phase is limited by rate of substrate transport to that phase. Since the amount of growth at the interface and substrate transport to the continuous phase depend on the interfacial area, two limiting cases for the decrease of interfacial area with substrate consumption are also considered in this investigation. The first and third models are compared with available experimental data.     

Growth models of cultures with two liquid phases. III. Continuous cultures

Some mathematical models, which have been used to describe batch growth in fermentations with two liquid phases present, are used to predict the behavior of continuous fermentations in a chemostat. Two types of dispersed systems are considered in this investigation. In the first, type, it is assumed that the composition of the dispersed phase is such that, increased substrate utilization results in a decreased substrate concentration with no change in the interfacial area. In the second type of system, the dispersed phase is assumed to be pure substrate; therefore, the substrate concentration in the dispersed phase remains constant but the interfacial area is affected by changes in dilution rate. Three special cases are examined for each type of system in order to examine the effect of the interfacial area, the phase equilibrium constant, and the mass transfer coefficient on system performance. Comparison of two of the models with available experimental data shows fair agreement, between model and data.     

The kinetics of yeast growth on pure hydrocarbons

The kinetics of C. tropicalis growth were investigated with pure n-hexadecane as dispersed phase substrate. Two distinct growth phases were found: In the first phase, exponential growth was independent of stirrer speed. The onset of the second phase, one of linear growth, was determined by stirrer speed. By the use of two different fermenter types, it was shown that the drop size of the dispersed phase was not primarily responsible for the observed kinetics. It was considered that the formation of biological flocs determined the observed growth pattern. This was substantiated by the results of continuous cultures in the different fermenter types, with various substrate concentrations.     

Growth models of cultures with two liquid phases. V. Substrate dissolved in dispersed phase—experimental observations

The effects of inoculum size, dispersed phase volume and substrate concentration on the batch growth of Candida lipolytica are investigated in a model system composed of n-hexadecane dissolved in dewaxed gas oil. Tabular values and parameters are presented for 16 different experiments. All of the batch growth curves exhibited a linear growth region with the length of the region ranging from 1.5 to 9.5 hours. The rate of linear growth varied both with change in dispersed phase volume and initial dispersed phase substrate concentration. A qualitative analysis of the results is presented and possible explanations for the observed linear growth rates are discussed.     

Design and physical characteristics of a multistage,continuous tower fermentor

A multistage tower laboratory fermentor has been constructed consisting of eight compartments separated by sieve plates. Flow of substrate and air is concurrent from the bottom to the top of the column. It, was hoped that this system could be used to reproduce, simultaneously on a continuous basis, eight distinct phases of a batch growth curve. It was believed that the extent of batch curve simulation would depend upon the character of hydraulic mean residence time of broth in the column and in the individual compartments. The expected relationship did not occur. Rather it was found that growth in the column involved residence time characteristics not only for the fluid but also for the microorganisms, and for the growth limiting substrate. Depending upon the column operation, these could be distinct and different. The purpose of this investigation was to study the residence time distribution (RTD) of the continous (fluid) and dispersed (microorganisms) phases for model systems as well as for a yeast fermentation. Various degrees of flow nonideality, i.e., fluid blackflow and dispersed phase sedimentation, were noticed. The former seems to be due to interaction of the concurrent gas and liquid flow; it is particularly dependent upon void area of the sieve plate holes. Sedimentation is probably a function of plate design as well as cell size and density. It wa concluded that for a particular plate design the gas hold-up wass controlled by superficial air velocity and was the main parameter governing the differences between dispersed and continous phase(Rt1). This conclusion was supported by a computeraided styudy utilizing a mathematical model of fluid flow to fit the growth kinetics and cell distribution observed experimentally throughout the fermentor. Some advantages of foam control in the tower fermentor by surface active compounds are mentioned. Also, suggestions are made for carrying out fermentations that have two liquid phases, such as a hydrocarbon fermentation. The possibility of closely approximating plug-flow conditions in the multistage tower fermentor, a necessary condition for batch growth simulation, is discussed from a practical point of view.     

Combined discrete particle and continuum model predicting solid-state fermentation in a drum fermentor

The development of mathematical models facilitates industrial (large-scale) application of solid-state fermentation (SSF). In this study, a two-phase model of a drum fermentor is developed that consists of a discrete particle model (solid phase) and a continuum model (gas phase). The continuum model describes the distribution of air in the bed injected via an aeration pipe. The discrete particle model describes the solid phase. In previous work, mixing during SSF was predicted with the discrete particle model, although mixing simulations were not carried out in the current work. Heat and mass transfer between the two phases and biomass growth were implemented in the two-phase model. Validation experiments were conducted in a 28-dm3 drum fermentor. In this fermentor, sufficient aeration was provided to control the temperatures near the optimum value for growth during the first 45-50 hours. Several simulations were also conducted for different fermentor scales. Forced aeration via a single pipe in the drum fermentors did not provide homogeneous cooling in the substrate bed. Due to large temperature gradients, biomass yield decreased severely with increasing size of the fermentor. Improvement of air distribution would be required to avoid the need for frequent mixing events, during which growth is hampered. From these results, it was concluded that the two-phase model developed is a powerful tool to investigate design and scale-up of aerated (mixed) SSF fermentors.     

Division Cycle of Myxococcus xanthus III. Kinetics of Cell Growth and Protein Synthesis

The kinetics of cell growth and protein synthesis during the division cycle of Myxococcus xanthus was determined. The distribution of cell size for both septated and nonseptated bacteria was obtained by direct measurement of the lengths of 8,000 cells. The Collins-Richmond equation was modified to consider bacterial growth in two phases: growth and division. From the derived equation, the growth rate of individual cells was computed as a function of size. Nondividing cells (growth phase) comprised 91% of the population and took up 87% of the time of the division cycle. The absolute and specific growth rates of nondividing cells were observed to increase continually throughout the growth phase; the growth rate of dividing cells could not be determined accurately by this technique because of changes in the geometry of cells between the time of septation and physical separation. The rate of protein synthesis during the division cycle was measured by pulselabeling an exponential-phase culture with radio-active valine or arginine and then preparing the cells for quantitative autoradiography. By measuring the size of individual cells as well as the number of grains, the rate of protein synthesis as a function of cell size was obtained. Nondividing cells showed an increase in both the absolute and specific rates of protein synthesis throughout the growth phase; the specific rate of protein synthesis for dividing cells was low when compared to growthphase cells. Cell growth and protein synthesis are compared to the previously reported kinetics of deoxyribonucleic acid and ribonucleic acid synthesis during the division cycle.     

Metabolic flux model for an anchorage-dependent MDCK cell line: characteristic growth phases and minimum substrate consumption flux distribution

Up to now cell-culture based vaccine production processes only reach low productivities. The reasons are: (i) slow cell growth and (ii) low cell concentrations. To address these shortcomings, a quantitative analysis of the process conditions, especially the cell growth and the metabolic capabilities of the host cell line is required. For this purpose a MDCK cell based influenza vaccine production process was investigated. With a segregated growth model four distinct cell growth phases are distinguished in the batch process. In the first phase the cells attach to the surface of the microcarriers and show low metabolic activity. The second phase is characterized by exponential cell growth. In the third phase, preceded by a change in oxygen consumption, contact inhibition leads to a decrease in cell growth. Finally, the last phase before infection shows no further increase in cell numbers. To gain insight into the metabolic activity during these phases, a detailed metabolic model of MDCK cell was developed based on genome information and experimental analysis. The MDCK model was also used to calculate a theoretical flux distribution representing an optimized cell that only consumes a minimum of carbon sources. Comparing this minimum substrate consumption flux distribution to the fluxes estimated from experiments unveiled high overflow metabolism under the applied process conditions.     

Growth models of cultures with two liquid phases. 8. Experimental observations on droplet size and interfacial area

Because of the importance of the drop she distribution and interfacial area of the dispersed liquid phase in hydrocarbon fermentations, experiments were carried out to determine the drop size distribution and the interfacial area during batch fermentations of Candida lipolytica on gas oil and on n-hexadecane dissolved in dewaxed gas oil. The effects of cell concentration and dispersed phase volume fraction on size distribution and interfacial area were investigated. Measurements of interfacial tensions, densities, viscosities, and fatty acid concentrations were also made. The results show that the size distribution is skewed and that the Sauter mean diameter is in the range of 10 to 30 μ. Both the Sauter mean diameter and the interfacial area increased during the course of a batch fermentation; however, they decreased at the end of the fermentation. The interfacial area also increased with inoculum size.     

The embryonic and postnatal growth of rat and mouse. III. growth of the whole animal in the puberty, adult, and senescence phases in two inbred mouse strains (cpb-s and kba/2). Exponential growth, sudden changes in the growth rate, and a model for the regulation of the mitotic rate.

1. The growth model forumlated for prenatal and postnatal growth up to the middle of the puberty phase seems to be valid for the later postnatal phases as well, including adulthood and senescence. 2. In this model, growth consists of phases with exponential growth (constant specific growth rate) separated by sudden changes in the rate. 3. In the period described here, 7 phases can be distinguished, beginning with puberty (phase IV) and ending with senescence (phase X). 4. In 4 of these phases the growth rate does not differ significantly between the four groups of mice used. In the other 3 phases there are no differences between three of the groups but the fourth, one of the sexes of the CPB-S strain, differs very definitely from the rest. 5. Absence of growth occurred in some phases in one of the groups. 6. Some of the phases do not occur in all individual animals; this holds especially for phase VIII, which occurred in only about 25% of the animals. 7. The 'growth constants' postulated in Part I and now studied in the individual mice, tend to have a value of about 2. A theoretical model is described for the regulation of the mitotic rate giving 'growth constants' of about the same numerical value.     

A new combined differential-discrete cellular automaton approach for biofilm modeling: application for growth in gel beads

The theoretical basis and quantitative evaluation of a new approach for modeling biofilm growth are presented here. Soluble components (e.g., substrates) are represented in a continuous field, whereas discrete mapping is used for solid components (e.g., biomass). The spatial distribution of substrate is calculated by applying relaxation methods to the reaction-diffusion mass balance. A biomass density map is determined from direct integration in each grid cell of a substrate-limited growth equation. Spreading and distribution of biomass is modeled by a discrete cellular automaton algorithm. The ability of this model to represent diffusion-reaction-microbial growth systems was tested for a well-characterized system: immobilized cells growing in spherical gel beads. Good quantitative agreement with data for global oxygen consumption rate was found. The calculated concentration profiles of substrate and biomass in gel beads corresponded to those measured. Moreover, it was possible, using the discrete spreading algorithm, to predict the spatial two- and three-dimensional distribution of microorganisms in relation to, for example, substrate flux and inoculation density. The new technique looks promising for modeling diffusion-reaction-microbial growth processes in heterogeneous systems as they occur in biofilms.     

Growth models of cultures with two liquid phases. VI. Parameter estimation and statistical analysis

Parameter estimation studies have been conducted employing mathematical models developed previously by the investigators and experimental data collected by the last author. A batch fermentation process in which Candida lipolytica were cultured on n-hexadecane dissolved in dewaxed gas oil was employed to obtain the experimental data. The kinetic data from a number of batch experiments conducted at different initial substrate concentrations and different dispersed phase volume fractions were analyzed assuming that, the basic model parameters (maximum specific growth rate, saturation constant, substrate phase equilibrium constant, adsorption constant, desorption constant, etc.) did not change from experiment to experiment. The Gauss-Newton method with modification by Greenstadt, Eisenpress, Bard, and Carroll was used to minimize the conventional sum of squares criterion on the IBM 300/50 computer. The individual confidence intervals were obtained for each individual parameter. Tin- models were compared employing the F-test for equality of variances and an analysis of residuals. For the two best models, the estimated parameter values were compared with available experimental information. The results showed good agreement between the experimental data and the values predicted by the mathematical models. The results presented in this work did suggest that growth on small segregated drops may be more important than continuous phase growth on dissolved substrate.     

Surface-limited growth: a model for the synchronization of a growing bacterial culture through periodic starvation

This article analyses the Surface-Limited Growth Model put forward to explain the very tight synchrony, over more than ten division cycles, obtained experimentally by subjecting a growing bacterial culture to alternating periods of starvation and dilution, using inorganic phosphate as the limiting substrate. The Model states that when an essential nutrient is in limited supply, the rate of growth of an individual cell will be proportional to its surface area (and the current concentration of the limiting substance) rather than to its volume. This decrease in dimensionality from volume to surface is expected to favor the smaller cells and so result ultimately in a narrower size distribution. The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions, and its predictions depend on how the cell replication cycle is assumed to behave under these same circumstances. Two alternatives are considered: the volume at which cells divide is the same during the starvation phase as during steady-state exponential growth, and the cells adjust immediately to the changing growth rate. In the latter case, we have tested both C + D constant with time and C + D variable (where C + D is the time between initiation of chromosome replication and the corresponding cell division), the incremental value at any instant being computed separately for each individual cell from its current effective growth rate. The simulation results are of two sorts depending on the auxiliary assumptions used. Either the dilution-starvation cycles have no effect whatsoever on the cell volume distribution, or the width of the distribution decreases gradually with time, approaching zero slowly and asymptotically, but the mean cell volume decreases as well--directly contradicting experimental observations. We conclude that the Surface-Limited Growth Model is incapable of explaining the synchronization of cells by periodic starvation of a growing bacterial culture.     

Cell surface energy, contact angles and phase partition. I. Lymphocytic cell lines in biphasic aqueous mixtures

The surface energy of cells is the quantity which dominates certain physical interactions of cells such as adhesion to hydrophobic surfaces and phagocytosis. A linear relationship is derived relating the equilibrium constant obtained from phase partition in liquid-liquid systems with the surface energy difference obtained from contact-angle measurements. Using biphasic mixtures of Dextran and poly(ethylene glycol) in a medium of constant salt composition the expression is confirmed for transformed lymphocytic cell lines. The results demonstrate the importance of van der Waals' interactions in the phase-partition process, that phase partition can be used as a direct measure of cell surface hydrophobicity, and that the equilibrium constant of phase partition is directly related to the difference in the surface energy of the partitioned particle between the two phases.     

Dispersions in hydrocarbon fermentation. A retrospective study

A nondimensionalized plot, obtained by normalizing the drop-size distribution in the hydrocarbon phase using the Sauter mean diameter, shows a tendency towards self-preservation of the distribution. Changes of distribution in time during the course of fermentation, initial dispersed phase fraction, speed of rotation, and reactor size were taken into account. Using this self-preserving property, an empirical (single parameter) equation has been proposed for drop-size distribution. Data, available from the literature, are presented for non-biological and biological systems (gas-oil, n-hexadecane, and n-hexadecane dissolved in dewaxed gas oil as dispersed phases). The parameter, Sauter mean diameter, has been correlated with the operating conditions, and a critical review presented. Cell density was found to have significant effect on Sauter mean diameter. This effect has also been empirically explained. The possibilities of using generalized distribution in predicting the performance of fermenters is outlined.     

Alginate hydrogel microspheres and microcapsules prepared by spinning disk atomization

Our spinning disk atomization (SDA) can, relative to other existing techniques, produce micron-sized particles with very narrow size distribution. The aim of this work is to present this technology for the production of alginate microspheres and microcapsules. We atomized and gelled aqueous alginate solutions into very narrowly dispersed microspheres with sizes ranging from 300 to 600 microm. Here, the interest is to produce, at high rate, particles of a given size with a narrow size distribution and also to show a new method of encapsulation using SDA. The viscosity and flow rate contributions in the drop formation is qualitatively analyzed to show how they affect droplet size. In addition, a technique for high degree of encapsulation is presented in which yeast is used as a model system. The production of yeast-loaded microspheres by SDA shows the potential of the technique for biotechnology applications.     

Identification of Growth Phases and Influencing Factors in Cultivations with AGE1.HN Cells Using Set-Based Methods

Production of bio-pharmaceuticals in cell culture, such as mammalian cells, is challenging. Mathematical models can provide support to the analysis, optimization, and the operation of production processes. In particular, unstructured models are suited for these purposes, since they can be tailored to particular process conditions. To this end, growth phases and the most relevant factors influencing cell growth and product formation have to be identified. Due to noisy and erroneous experimental data, unknown kinetic parameters, and the large number of combinations of influencing factors, currently there are only limited structured approaches to tackle these issues. We outline a structured set-based approach to identify different growth phases and the factors influencing cell growth and metabolism. To this end, measurement uncertainties are taken explicitly into account to bound the time-dependent specific growth rate based on the observed increase of the cell concentration. Based on the bounds on the specific growth rate, we can identify qualitatively different growth phases and (in-)validate hypotheses on the factors influencing cell growth and metabolism. We apply the approach to a mammalian suspension cell line (AGE1.HN). We show that growth in batch culture can be divided into two main growth phases. The initial phase is characterized by exponential growth dynamics, which can be described consistently by a relatively simple unstructured and segregated model. The subsequent phase is characterized by a decrease in the specific growth rate, which, as shown, results from substrate limitation and the pH of the medium. An extended model is provided which describes the observed dynamics of cell growth and main metabolites, and the corresponding kinetic parameters as well as their confidence intervals are estimated. The study is complemented by an uncertainty and outlier analysis. Overall, we demonstrate utility of set-based methods for analyzing cell growth and metabolism under conditions of uncertainty.     

Growth of yeast on n-alkanes. II. Batch experiments

In this paper the results of the Monte Carlo simulations as described in an earlier paper are compared with those of batch experiments. A number of batch experiments were carried out at a low inoculation rate so that only a fraction of the oil drops were inoculated. Under these conditions the effect of the segregation of the oil phase is more clearly demonstrated. Special attention is paid to the preparation of actively growing yeast cells with which the cultures is inoculated. Also a method is developed to estimate the amount of actively growing cells with which the culture is inoculated. The other parameters necessary for the Monte Carlo simulation are measured in separate experiments: the maximum growth rate of the cells, oil drop size, and the drop parameters. Finally the growth curves (measured in the batch experiments) are compared with those calculated with the Monte Carlo procedure. A good agreement is found.